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First-order logic

First-order logic
A theory about some topic is usually first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions which map from that domain into it, finitely many predicates defined on that domain, and a recursive set of axioms which are believed to hold for those things. Sometimes "theory" is understood in a more formal sense, which is just a set of sentences in first-order logic. The adjective "first-order" distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.[1] In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets. First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Introduction[edit] . x in . Related:  The problems with philosophy

Model theory This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang and Keisler (1990):[1] universal algebra + logic = model theory. Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997): although model theorists are also interested in the study of fields. In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. Branches of model theory[edit] This article focuses on finitary first order model theory of infinite structures. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. and or

Signature (logic) In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. Signatures play the same role in mathematics as type signatures in computer programming. They are rarely made explicit in more philosophical treatments of logic. Formally, a (single-sorted) signature can be defined as a triple σ = (Sfunc, Srel, ar), where Sfunc and Srel are disjoint sets not containing any other basic logical symbols, called respectively function symbols (examples: +, ×, 0, 1) andrelation symbols or predicates (examples: ≤, ∈), and a function ar: Sfunc Srel → which assigns a non-negative integer called arity to every function or relation symbol. A signature with no function symbols is called a relational signature, and a signature with no relation symbols is called an algebraic signature. Symbol types S.

Quantification In logic, quantification is the binding of a variable ranging over a domain of discourse. The variable thereby becomes bound by an operator called a quantifier. Academic discussion of quantification refers more often to this meaning of the term than the preceding one. Natural language[edit] All known human languages make use of quantification (Wiese 2004). Every glass in my recent order was chipped.Some of the people standing across the river have white armbands.Most of the people I talked to didn't have a clue who the candidates were.A lot of people are smart. The words in italics are quantifiers. The study of quantification in natural languages is much more difficult than the corresponding problem for formal languages. Montague grammar gives a novel formal semantics of natural languages. Logic[edit] In language and logic, quantification is a construct that specifies the quantity of specimens in the domain of discourse that apply to (or satisfy) an open formula. Mathematics[edit] . means

Hydrogen Hydrogen is a chemical element with chemical symbol H and atomic number 1. With a standard atomic weight of circa 7000100800000000000♠1.008, hydrogen is the lightest element on the periodic table. Its monatomic form (H) is the most abundant chemical substance in the Universe, constituting roughly 75% of all baryonic mass.[10][note 1] Non-remnant stars are mainly composed of hydrogen in the plasma state. Hydrogen gas was first artificially produced in the early 16th century by the reaction of acids on metals. Industrial production is mainly from steam reforming natural gas, and less often from more energy-intensive methods such as the electrolysis of water.[13] Most hydrogen is used near the site of its production, the two largest uses being fossil fuel processing (e.g., hydrocracking) and ammonia production, mostly for the fertilizer market. Properties Combustion Explosion of a hydrogen–air mixture. 2 H2(g) + O2(g) → 2 H2O(l) + 572 kJ (286 kJ/mol)[note 2] Electron energy levels Phases Notes

COST-ARTS COST-TUD1102 STSM : Call for STSM Applications for Researchers . . . 2015 is the last year of the Action's operation and we have a busy schedule. As events are fixed in the calendar, we will be detailing them in the 2015 Event page 2015: ISSUE 1 Featuring Final Conference 2nd ARTS Competition Eastern/Western Europe Dissemination Workshops ARTS Vocabulary 2014: ISSUE 4 Featuring Reach Out with the Wider Community Recent Events Future Event: Meeting in November Next Year Ideas 2014: ISSUE 3 Featuring The call for participation in the 2nd Summer School Autonomic Road Transport Support Systems: Behavioural Responses & Impact 2014: ISSUE 2 Featuring REPORT OF Annual Meeting at Lisbon, February 27/28 STSMs – short term visits Training School THIS YEAR Wiki for the COST ARTS Vocabulary Progress on COST ARTS Book Featuring the COST-ARTS road map.

Operator grammar Operator grammar is a mathematical theory of human language that explains how language carries information. This theory is the culmination of the life work of Zellig Harris, with major publications toward the end of the last century. Operator Grammar proposes that each human language is a self-organizing system in which both the syntactic and semantic properties of a word are established purely in relation to other words. Thus, no external system (metalanguage) is required to define the rules of a language. Dependency[edit] In each language the dependency relation among words gives rise to syntactic categories in which the allowable arguments of an operator are defined in terms of their dependency requirements. The categories in operator grammar are universal and are defined purely in terms of how words relate to other words, and do not rely on an external set of categories such as noun, verb, adjective, adverb, preposition, conjunction, etc. Likelihood[edit] Reduction[edit]

Cartesian product Cartesian product of the sets and The simplest case of a Cartesian product is the Cartesian square, which returns a set from two sets. A Cartesian product of n sets can be represented by an array of n dimensions, where each element is an n-tuple. The Cartesian product is named after René Descartes,[1] whose formulation of analytic geometry gave rise to the concept. Examples[edit] A deck of cards[edit] An illustrative example is the standard 52-card deck. Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}. Suits × Ranks returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}. A two-dimensional coordinate system[edit] An example in analytic geometry is the Cartesian plane. Most common implementation (set theory)[edit] A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. . , where For example: Similarly

Interpretation Interpretation may refer to: Culture[edit] Law[edit] Authentic interpretation, the official interpretation of a statute issued by the statute's legislatorFinancial Accounting Standards Board Interpretations, part of the United States Generally Accepted Accounting Principles (US GAAP)Interpretation Act, a stock short title used for legislation relating to interpretation of legislationJudicial interpretation, an interpretation of law by a judiciaryStatutory interpretation, determining the meaning of legislation Math and computing[edit] Media[edit] Neuroscience[edit] Philosophy[edit] Religion[edit] See also[edit] What is a homunculus and what does it tell scientists A homunculus is a sensory map of your body, so it looks like an oddly proportioned human. The reason it's oddly proportioned is that a homunculus represents each part of the body in proportion to its number of sensory neural connections and not its actual size. The layout of the sensory neural connections throughout your body determines the level of sensitivity each area of your body has, so the hands on a sensory homunculus are its largest body parts, exaggerated to an almost comical degree, while the arms are quite skinny. The homunculus, then, gives a vivid picture of where our sensory system gets the most bang for its buck. Chemoreceptors, which sense chemicals.

IPC 2014 The international planning competition is a (nearly) biennial event organized in the context of the International Conference on Planning and Scheduling (ICAPS). The competition has different goals, including, providing an empirical comparison of the state of the art of planning systems, highlighting challenges to the Planning community, proposing new directions for research and new links with other fields of AI, and providing new data sets to be used by the research community as benchmarks. How to reference the competition? Vallati, M. and Chrpa, L. and Grzes, M. and McCluskey, T.L. and Roberts, M. and Sanner, S. (2015) The 2014 International Planning Competition: Progress and Trends , AI Magazine. ISSN 0738-4602. General Information Development Environment System Info Development Environment System WIKI DES ticketing system Mailing list Important dates (Preliminar) Post-IPC Organizers Previous competitions Sponsors The Queensgate Grid The University of Huddersfield

Part of speech Controversies[edit] Linguists recognize that the above list of eight word classes is drastically simplified and artificial.[2] For example, "adverb" is to some extent a catch-all class that includes words with many different functions. Some have even argued that the most basic of category distinctions, that of nouns and verbs, is unfounded,[3] or not applicable to certain languages.[4] English[edit] A diagram of English categories in accordance with modern linguistic studies English words have been traditionally classified into eight lexical categories, or parts of speech (and are still done so in most dictionaries): Noun any abstract or concrete entity; a person (police officer, Michael), place (coastline, London), thing (necktie, television), idea (happiness), or quality (bravery) Pronoun any substitute for a noun or noun phrase Adjective any qualifier of a noun Verb any action (walk), occurrence (happen), or state of being (be) Adverb Preposition any establisher of relation and syntactic context